Statement
If x1, x2,..., xd are d complex numbers that are linearly independent over the rational numbers, and y1, y2,...,yl are l complex numbers that are also linearly independent over the rational numbers, and if dl > d + l, then at least one of the following dl numbers is transcendental:
The most interesting case is when d = 3 and l = 2, in which case there are six exponentials, hence the name of the result. The theorem is weaker than the related but thus far unproved four exponentials conjecture, whereby the strict inequality dl > d + l is replaced with dl ≥ d + l, thus allowing d = l = 2.
The theorem can be stated in terms of logarithms by introducing the set L of logarithms of algebraic numbers:
The theorem then says that if λij are elements of L for i = 1, 2 and j = 1, 2, 3, such that λ11, λ12, and λ13 are linearly independent over the rational numbers, and λ11 and λ21 are also linearly independent over the rational numbers, then the matrix
has rank 2.
Read more about this topic: Six Exponentials Theorem
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